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Let X be a C1+ vector field on a compact Riemannian manifold M with dimension d ≥ 3. Let Λ be a transitive singular hypebolic set with positive volume. We show that Λ = M and Λ is a uniformly hyperbolic set without singularities.

There are two objects naturally associated with a braid $\beta\in B_n$ of pseudo-Anosov type: a (relative) pseudo-Anosov homeomorphism $\varphi_\beta\colon S^2\to S^2$; and the finite volume complete hyperbolic structure on the 3-manifold $M_\beta$ obtained by excising the braid closure of $\beta$, together with its braid axis, from $S^3$. We show the disconnect between these objects, by exhibiting a family of braids $\{\beta_q:q\in{\mathbb{Q}}\cap(0,1/3]\}$ with the properties that: on the one hand, there is a fixed homeomorphism $\varphi_0\colon S^2\to S^2$ to which the (suitably normalized) homeomorphisms $\varphi_{\beta_{q}}$ converge as $q\to 0$; while on the other hand, there are infinitely many distinct hyperbolic 3-manifolds which arise as geometric limits of the form $\lim_{k\to\infty} M_{\beta_{q_k}}$, for sequences $q_k\to 0$., Comment: Author accepted manuscript

We use entropy theory as a new tool to study sectional hyperbolic flows in any dimension. We show that for C 1 flows, every sectional hyperbolic set Λ is entropy expansive, and the topological entropy varies continuously with the flow. Furthermore, if Λ is Lyapunov stable, then it has positive entropy; in addition, if Λ is a chain recurrent class, then it contains a periodic orbit. As a corollary, we prove that for C 1 generic flows, every Lorenz-like class is an attractor.

Let $\Delta$ be a hyperbolic triangle with a fixed area $\varphi$. We prove that for all but countably many $\varphi$, generic choices of $\Delta$ have the property that the group generated by the $\pi$--rotations about the midpoints of the sides of the triangle admits no nontrivial relations. By contrast, we show for all $\varphi\in(0,\pi)\setminus\mathbb{Q}\pi$, a dense set of triangles does afford nontrivial relations, which in the generic case map to hyperbolic translations. To establish this fact, we study the deformation space $\mathfrak{C}_\theta$ of singular hyperbolic metrics on a torus with a single cone point of angle $\theta=2(\pi-\varphi)$, and answer an analogous question for the holonomy map $\rho_\xi$ of such a hyperbolic structure $\xi$. In an appendix by X.~Gao, concrete examples of $\theta$ and $\xi\in\mathfrak{C}_\theta$ are given where the image of each $\rho_\xi$ is finitely presented, non-free and torsion-free; in fact, those images will be isomorphic to the fundamental groups of closed hyperbolic 3--manifolds., Comment: 32 pages. To appear in Math. Z

We consider an autonomous system admitting an invariant manifold $\mathcal{M}$. The following questions are discussed: (i) what are the conditions ensuring exponential stability of the invariant manifold? (ii) does every motion attracting by $\mathcal{M}$ tend to some motion on $\mathcal{M}$ (i.e. have an asymptotic phase)? (iii) what is the geometrical structure of the set formed by orbits approaching a given orbit? We get an answer to (i) in terms of Lyapunov functions omitting the assumption that the normal bundle of $\mathcal{M}$ is trivial. An affirmative answer to (ii) is obtained for invariant manifold $\mathcal{M}$ with partially hyperbolic structure of tangent bundle. In this case, the existence of asymptotic phase is obtained under new conditions involving contraction rates of the linearized flow in normal and tangential to $\mathcal{M}$ directions. To answer the question (iii), we show that a neighborhood of $\mathcal{M}$ has a structure of invariant foliation each leaf of which corresponds to motions with common asymptotic phase. In contrast to theory of cascades, our technique exploits the classical Lyapunov–Perron method of integral equations.

Given the fundamental group $\Gamma$ of a finite-volume complete hyperbolic $3$-manifold $M$, it is possible to associate to any representation $\rho:\Gamma \rightarrow \text{Isom}(\mathbb{H}^3)$ a numerical invariant called volume. This invariant is bounded by the hyperbolic volume of $M$ and satisfies a rigidity condition: if the volume of $\rho$ is maximal, then $\rho$ must be conjugated to the holonomy of the hyperbolic structure of $M$. This paper generalizes this rigidity result by showing that if a sequence of representations of $\Gamma$ into $\text{Isom}(\mathbb{H}^3)$ satisfies $\lim_{n \to \infty} \text{Vol}(\rho_n) = \text{Vol}(M)$, then there must exist a sequence of elements $g_n \in \text{Isom}(\mathbb{H}^3)$ such that the representations $g_n \circ \rho_n \circ g_n^{-1}$ converge to the holonomy of $M$. In particular if the sequence $\rho_n$ converges to an ideal point of the character variety, then the sequence of volumes must stay away from the maximum. We conclude by generalizing the result to the case of $k$-manifolds and representations in $\text{Isom}(\mathbb H^m)$, where $m\geq k$., Comment: 21 pages

Singular and sectional hyperbolic sets are the objects of the extension of the classical Smale Hyperbolic Theory to flows having invariant sets with singularities accumulated by regular orbits within the set. It is by now well-known that (partially) hyperbolic sets admit adapted metrics. We show the existence of singular adapted metrics for any singular hyperbolic set with respect to a $C^{1}$ vector field on finite dimensional compact manifolds. Moreover, we obtain 2-sectional adapted metrics for certain open classes of 2-sectional hyperbolic sets and also for any hyperbolic set., Comment: 23 pages, 1 figure. Improved the statements of the results and corrected the proof of the main theorem. To appear in Bull. Brazilian Mathematical Society

This paper is an extension and generalization of some previous works, such as the study of M. Benalili and A. Lansari. Indeed, these authors, in their work about the finite co-dimension ideals of Lie algebras of vector fields, restricted their study to fields $$X_0$$ X 0 of the form $$X_0=\sum _{i=1}^{n}( \alpha _i \cdot x_i+\beta _i\cdot x_i^{1+m_i}) \frac{\partial }{\partial x_i}$$ X 0 = ∑ i = 1 n ( α i · x i + β i · x i 1 + m i ) ∂ ∂ x i , where $$\alpha _i, \beta _i $$ α i , β i are positive and $$m_i$$ m i are even natural integers. We will first study the sub-algebra U of the Lie-Fréchet space E, containing vector fields of the form $$Y_0 = X_0^+ + X_0^- + Z_0$$ Y 0 = X 0 + + X 0 - + Z 0 , such as $$ X_0\left( x,y\right) =A\left( x,y\right) =\left( A^{-}\left( x \right) ,A^{+}\left( y\right) \right) $$ X 0 x , y = A x , y = A - x , A + y , with $$A^-$$ A - (respectively, $$ A^+ $$ A + ) a symmetric matrix having eigenvalues $$ \lambda < 0$$ λ < 0 (respectively, $$\lambda >0 $$ λ > 0 ) and $$Z_0$$ Z 0 are germs infinitely flat at the origin. This sub-algebra admits a hyperbolic structure for the diffeomorphism $$\psi _{t*}=(exp\cdot tY_0)_*$$ ψ t ∗ = ( e x p · t Y 0 ) ∗ . In a second step, we will show that the infinitesimal generator $$ad_{-X}$$ a d - X is an epimorphism of this admissible Lie sub-algebra U. We then deduce, by our fundamental lemma, that $$U=E$$ U = E .

—We consider an arbitrary C1 diffeomorphism f that acts from an open subset U of a Riemannian manifold M of dimension m, m ≥ 2, to f(U) ⊂ M. Let A be a compact f-invariant (i.e., f(A) = A) subset in U. We propose various sufficient conditions under which A is a hyperbolic set of f.

A blender is a hyperbolic set with a stable or unstable invariant manifold that behaves as a geometric object of a dimension larger than that of the respective manifold itself. Blenders have been constructed in diffeomorphisms with a phase space of dimension at least three. We consider here the question of how one can identify, characterize and also visualize the underlying hyperbolic set of a given diffeomorphism to verify whether it actually is a blender or not. More specifically, we employ advanced numerical techniques for the computation of global manifolds to identify the hyperbolic set and its stable and unstable manifolds in an explicit Henon-like family of three-dimensional diffeomorphisms. This allows to determine and illustrate whether the hyperbolic set is a blender; in particular, we consider as a distinguishing feature the self-similar structure of the intersection set of the respective global invariant manifold with a plane. By checking and illustrating a denseness property, we are able to identify a parameter range over which the hyperbolic set is a blender, and we discuss and illustrate how the blender disappears.

Hyperbolicity is a cornerstone of nonlinear dynamical systems theory. Hyperbolic dynamics are characterized by the presence of expanding and contracting directions for the derivative along the trajectories of the system. Hyperbolic dynamical systems enjoy many interesting properties like structural stability, ergodicity, transitivity, etc. In this letter, we describe a hybrid systems framework to compute invariant sets with a hyperbolic structure for a given dynamical system. The method relies on an abstraction (also known as symbolic image or bisimulation) of the state space of the system, and on path-complete ``Lyapunov-like'' techniques to compute the expanding and contracting directions for the derivative along the trajectories of the system. The method is illustrated on a numerical example: the Ikeda map for which an invariant set with hyperbolic structure is computed using the framework

In this paper, we study the manifold structure and the relative hyperbolic structure of right-angled Coxeter groups with planar nerves. We then apply our results to the quasi-isometry problem for this class of right-angled Coxeter groups.

It is known that the quantum SU ( 2 ) invariant of a closed 3 –manifold at q = exp ( 2 π − 1 ∕ N ) is of polynomial order as N → ∞ . Recently, Chen and Yang conjectured that the quantum SU ( 2 ) invariant of a closed hyperbolic 3 –manifold at q = exp ( 4 π − 1 ∕ N ) is of order exp ( N ⋅ ς ( M ) ) , where ς ( M ) is a normalized complex volume of M . We can regard this conjecture as a kind of “volume conjecture”, which is an important topic from the viewpoint that it relates quantum topology and hyperbolic geometry. In this paper, we give a concrete presentation of the asymptotic expansion of the quantum SU ( 2 ) invariant at q = exp ( 4 π − 1 ∕ N ) for closed hyperbolic 3 –manifolds obtained from the 3 –sphere by integral surgery along the figure-eight knot. In particular, the leading term of the expansion is exp ( N ⋅ ς ( M ) ) , which gives a proof of the Chen–Yang conjecture for such 3 –manifolds. Further, the semiclassical part of the expansion is a constant multiple of the square root of the Reidemeister torsion for such 3 –manifolds. We expect that the higher-order coefficients of the expansion would be “new” invariants, which are related to “quantization” of the hyperbolic structure of a closed hyperbolic 3 –manifold.

Hyperbolic graph convolutional networks (GCNs) demonstrate powerful representation ability to model graphs with hierarchical structure. Existing hyperbolic GCNs resort to tangent spaces to realize graph convolution on hyperbolic manifolds, which is inferior because tangent space is only a local approximation of a manifold. In this paper, we propose a hyperbolic-to-hyperbolic graph convolutional network (H2H-GCN) that directly works on hyperbolic manifolds. Specifically, we developed a manifold-preserving graph convolution that consists of a hyperbolic feature transformation and a hyperbolic neighborhood aggregation. The hyperbolic feature transformation works as linear transformation on hyperbolic manifolds. It ensures the transformed node representations still lie on the hyperbolic manifold by imposing the orthogonal constraint on the transformation sub-matrix. The hyperbolic neighborhood aggregation updates each node representation via the Einstein midpoint. The H2H-GCN avoids the distortion caused by tangent space approximations and keeps the global hyperbolic structure. Extensive experiments show that the H2H-GCN achieves substantial improvements on the link prediction, node classification, and graph classification tasks., CVPR2021, Oral

A vector field X is called a star flow if every periodic orbit, of any vector field C1-close to X, is hyperbolic. It is known that the chain recurrence classes of a generic star flow X on a 3 or 4 manifold are either hyperbolic or singular hyperbolic (see [MPP] for 3-manifolds and [GLW] on 4-manifolds). As it is defined, the notion of singular hyperbolicity forces the singularities in the same class to have the same index. However, in higher dimension (i.e $\geq 5$) \cite{BdL} shows that singularities of different indices may be robustly in the same chain recurrence class of a star flow. Therefore the usual notion of singular hyperbolicity is not enough for characterizing the star flows. We present a form of hyperbolicity (called multi-singular hyperbolic) which makes compatible the hyperbolic structure of regular orbits together with the one of singularities even if they have different indices. We show that multisingular hyperbolicity implies that the flow is star, and conversely, there is a C1-open and dense subset of the an open set of star flows which are multisingular hyperbolic. More generally, for most of the hyperbolic structures (dominated splitting, partial hyperbolicity etc...) well defined on regular orbits, we propose a way for generalizing it to a compact set containing singular points., There are new results in section 7 compared with the previous version

We show that every plumbing of disc bundles over surfaces whose genera satisfy a simple inequality may be embedded as a convex submanifold in some closed hyperbolic four-manifold. In particular its interior has a geometrically finite hyperbolic structure that covers a closed hyperbolic four-manifold., 18 pages, 11 figures

We describe the phase space structures related to the semi-major axis of Molniya-like satellites subject to tesseral and lunisolar resonances. In particular, the questions answered in this contribution are: (i) we study the indirect interplay of the critical inclination resonance on the semi-geosynchronous resonance using a hierarchy of more realistic dynamical systems, thus discussing the dynamics beyond the integrable approximation. By introducing ad hoc tractable models averaged over fast angles, (ii) we numerically demarcate the hyperbolic structures organising the long-term dynamics via Fast Lyapunov Indicators cartography. Based on the publicly available two-line elements space orbital data, (iii) we identify two satellites, namely Molniya 1-69 and Molniya 1-87, displaying fingerprints consistent with the dynamics associated to the hyperbolic set. Finally, (iv) the computations of their associated dynamical maps highlight that the spacecraft are trapped within the hyperbolic tangle. This research therefore reports evidence of actual artificial satellites in the near-Earth environment whose dynamics are ruled by manifolds and resonant mechanisms. The tools, formalism and methodologies we present are exportable to other region of space subject to similar commensurabilities as the geosynchronous region., Comment: 26 pages, 9 figures. Comments and feedback appreciated

Prior approaches for multimodal sentiment and emotion recognition (SER) exploit input data representations and neural networks based on the classical Euclidean geometry. Recently, however, the hyperbolic metric proved to be a powerful tool for data mapping, being able to capture the hierarchical structure of the relations among elements in the data. In this paper we propose the use of hyperbolic learning for SER, and show that the inclusion in the neural network of hyperbolic structures mapping the input into the hyperbolic space can improve the quality of the predictions. The benefits brought by the hyperbolic features are evaluated by developing extensions of existing methods following two approaches. From one side, we modified state-of-the-art models by including hyperbolic output layers. From the other, we generated hybrid neural network architectures by combining hyperbolic and Euclidean layers according to different schemes. The proposed hyperbolic models were tested on several classification tasks applied to benchmark multimodal SER datasets. Experiments gave strong evidence that in both simple and complex networks the introduction of a hyperbolic structure results in an improvement of the model accuracy. Specifically, the combined use of hyperbolic and Euclidean layers showed superior performance in almost all the classification tasks.

In this paper, a transmit digital multi-beam forming method based on hyperbolic fractional delay filter is presented, which provides effective time delay compensation and can reduce the consumption of hardware resources. In the proposed method,firstly, the hyperbolic function is expanded in the way of infinite product, and only two sub-filters are included in the expansion. Then, the hyperbolic fractional delay filter is obtained by using the solved sub-filter, so as to realize the clock fractional delay compensation. Then, the transmit digital multibeam forming is realized by using the designed fractional delay filter based on hyperbolic structure. Finally, the feasibility and effectiveness of the proposed design method are verified by experimental simulation.

Despite that Hyperbolic Metamaterial (HMM) has demonstrated sub-wavelength focusing inside of it, sub-wavelength imaging in free space of HMM is rarely introduced. The decay of hyperbolic momentum space outside the hyperbolic medium has hindered the realization of sub-wavelengh focusing in the near field of HMM. Furthermore, manipulating the negatively refracted waves exiting the HMM have addressed another major obstacle to realize free space sub-wavelength focusing. In this work, we report extended sub-wavelength focusing in free space based on negative refraction of light exiting the HMM. The proposed structure is composed of multilayers of doped InAs/intrinsic InAs integrated with metallic slit. We theoretically simulate the doped InAs/intrinsic InAs HMM and investigate the negative refraction behavior outside the HMM. We optimized the structure for achieving high resolution down to 0.2λ, extended to a distance of 3.2 µm in free space. Also, sub-wavelength focusing in free space has been studied at different doping concentrations showing that the small doping concentrations exhibit enhancement in resolution at short distances up to 600 nm away from the HMM. Extending the focusing distance is achieved up to distance 3.5 µm from the hyperbolic structure by manipulating the doping concentration. This proposed lens configuration is expected to find potential usage in mid IR thermal imaging and photolithography application.

Summary Patterns of gene expressions play a key role in determining cell state. Although correlations in gene expressions have been well documented, most of the current methods treat them as independent variables. One way to take into account gene correlations is to find a low-dimensional curved geometry that describes variation in the data. Here we develop such a method and find that gene expression across multiple cell types exhibits a low-dimensional hyperbolic structure. When more genes are taken into account, hyperbolic effects become stronger but representation remains low dimensional. The size of the hyperbolic map, which indicates the hierarchical depth of the data, was the largest for human cells, the smallest for mouse embryonic cells, and intermediate in differentiated cells from different mouse organs. We also describe how hyperbolic metric can be incorporated into the t-SNE method to improve visualizations compared with leading methods., Graphical abstract, Highlights • A method to identify underlying low-dimensional geometry in high-dimensional dataset • Gene expression data exhibit local Euclidean and large-scale hyperbolic geometry • The size of the hyperbolic map is larger in differentiated and brain cells • Taking into account hyperbolic geometry yields improved visualization of data, Genes; Cell Biology; Complex Systems

We prove that every singular hyperbolic chain transitive set with a singularity does not admit the shadowing property. Using this result we show that if a star flow has the shadowing property on its chain recurrent set then it satisfies Axiom A and the no-cycle conditions; and that if a multisingular hyperbolic set has the shadowing property then it is hyperbolic.

We prove that the closure of the closed orbits of a generic geodesic flow on a closed Riemannian n >= 2 dimensional manifold is a uniformly hyperbolic set if the shadowing property holds C2-robustly on the metric. We obtain analogous results using weak specification and the shadowing property allowing bounded time reparametrization., The authors were partially supported by the Project ‘New trends in Lyapunov exponents’ (PTDC/MAT-PUR/29126/2017). MB was partially supported by FCT - ‘Fundação para a Ciência e a Tecnologia’, through Centro de Matemática e Aplicações (CMA-UBI), Universidade da Beira Interior, project UID/MAT/00212/2013. JLD was partially supported by the Project CEMAPRE - UID/MULTI/00491/2019 financed by FCT/MCTES through national funds. MJT was partially supported by the Research Centre of Mathematics of the University of Minho with the Portuguese Funds from the Fundação para a Ciência e a Tecnologia, through the Project UID/MAT/00013/2013.

In our work we study the dipole placed in homogenized hyperbolic media surrounding by air. We investigate the enhancement of outcoupled radiation due to the mutual orientation of a dipole and a hyperbolic metamaterial face. We analyze the radiation pattern of different shapes of the hyperbolic structure in order to find the optimum geometrical parameters. These results lead to the efficient emission control and Purcell effect implementation.

A biperiodic alternating link has an alternating quotient link in the thickened torus. In this paper, we focus on semi-regular links, a class of biperiodic alternating links whose hyperbolic structure can be immediately determined from a corresponding Euclidean tiling. Consequently, we determine the exact volumes of semi-regular links. We relate their commensurability and arithmeticity to the corresponding tiling, and assuming a conjecture of Milnor, we show there exist infinitely many pairwise incommensurable semi-regular links with the same invariant trace field. We show that only two semi-regular links have totally geodesic checkerboard surfaces; these two links satisfy the Volume Density Conjecture. Finally, we give conditions implying that many additional biperiodic alternating links are hyperbolic and admit a positively oriented, unimodular geometric triangulation. We also provide sharp upper and lower volume bounds for these links., 25 pages, 15 figures. V2: Minor changes. Added reference, fixed typos, and clarified proof of Theorem 5.1

Let Λ be a random compact invariant (with respect to a random map) set with a uniform tangent bundle splitting. We show that if there is a random non-uniformly hyperbolic variable x0 on Λ and the space of limit points of time averages of random Dirac measures on the orbit of x0 coincides with the space of invariant probability measures supported on Λ then, Λ is a random uniformly hyperbolic set.

Let $S_g$ ($g\geq 2$) be a closed surface of genus $g$. Let $K$ be any real number field and $A$ be any quaternion algebra over $K$ such that $A\otimes_K\mathbb{R}\cong M_2(\mathbb{R})$. We show that there exists a hyperbolic structure on $S_g$ such that $K$ and $A$ arise as its invariant trace field and invariant quaternion algebra.

In this paper we investigate the quasi-shadowing property for C1 random dynamical systems on their random partially hyperbolic sets. It is shown that for any pseudo orbit {xk} −∞ +∞ on a random partially hyperbolic set there exists a “center” pseudo orbit {yk} −∞ +∞ shadowing it in the sense that yk+1 is obtained from the image of yk by a motion along the center direction. Moreover, when the random partially hyperbolic set has a local product structure, the above “center” pseudo orbit {yk} −∞ +∞ can be chosen such that yk+1 and the image of yk lie in their common center leaf.

Schottky groups are exactly those Kleinian groups providing the regular lowest planar uniformizations of closed Riemann surfaces and also the ones providing to the interior of a handlebody of a complete hyperbolic structure with injectivity radius bounded away from zero. The space parametrizing quasiconformal deformations of Schottky groups of a fixed rank $g \geq 1$ is the marked Schottky space ${\mathcal M}{\mathcal S}_{g}$; this being a complex manifold of dimension $3(g-1)$ for $g \geq 2$ and being isomorphic to the punctured unit disc for $g=1$. In this paper we provide a complete description of the real structures of ${\mathcal M}{\mathcal S}_{g}$, up to holomorphic automorphisms, together their real part.

It is known that a knot complement can be decomposed into ideal octahedra along a knot diagram. A solution to the gluing equations applied to this decomposition gives a pseudo-developing map of the knot complement, which will be called a pseudo-hyperbolic structure. In this paper, we study these in terms of segment and region variables which are motivated by the volume conjecture so that we can compute complex volumes of all the boundary parabolic representations explicitly. We investigate the octahedral developing and holonomy representation carefully, and obtain a concrete formula of Wirtinger generators for the representation and also cusp shape. We demonstrate explicit solutions for $T(2,N)$ torus knots, $J(N,M)$ knots and also for other interesting knots as examples. Using these solutions we can observe the asymptotic behavior of complex volumes and cusp shapes of these knots. We note that this construction works for any knot or link, and reflects systematically both geometric properties of the knot complement and combinatorial aspect of the knot diagram., 55 pages, 31 figures

In this paper we present in a topological way the construction of the orientable surface with only one end and infinite genus, called \emph{The Infinite Loch Ness Monster}. In fact, we introduce a flat and hyperbolic construction of this surface. We discuss how the name of this surface has evolved and how it has been historically understood.

Given a reduced alternating diagram for a link, we obtain conditions that guarantee that the link complement has a complete hyperbolic structure, crossing arcs are the edges of an ideal geodesic triangulation, and every crossing arc is isotopic to a simple geodesic. The latter was conjectured by Sakuma and Weeks in 1995. We provide infinite families of closed braids for which our conditions hold., To appear in Asian Journal of Matematics. 18 pages, 12 figures

This paper focuses on generic properties of continuous dynamical systems. We prove \begin{document}$C^1$\end{document} weak Palis conjecture for nonsingular flows: Morse-Smale vector fields and vector fields admitting horseshoes are open and dense among \begin{document}$C^1$\end{document} nonsingular vector fields. Our arguments contain three main ingredients: linear Poincare flow, Liao's selecting lemma and the adapting of Crovisier's central model. Firstly, by studying the linear Poincare flow, we prove for a \begin{document}$C^1$\end{document} generic vector field away from horseshoes, any non-trivial nonsingular chain recurrent class contains a minimal set which is partially hyperbolic with 1-dimensional center with respect to the linear Poincare flow. Secondly, to understand the neutral behaviour of the 1-dimensional center, we adapt Crovisier's central model. The difficulties are that we can not build invariant plaque family of any time, the periodic point of a flow is not periodic for the discrete time map. Through delicate analysis of the center manifold of a periodic orbit near the partially hyperbolic set, we manage to yield nice periodic points such that their stable manifolds and unstable manifolds are well-placed for transverse intersection.

We consider a discrete dynamical system on a pseudo-Riemannian manifold and we determine the concept of a hyperbolic set for it. We insert a condition in the definition of a hyperbolic set which implies to the unique decomposition of a part of tangent space (at each point of this set) to two unstable and stable subspaces with exponentially increasing and exponentially decreasing dynamics on them. We prove the continuity of this decomposition via the metric created by a torsion-free pseudo-Riemannian connection. We present a global attractor for a diffeomorphism on an open submanifold of the hyperbolic space \begin{document}$H^2(1)$\end{document} which is not a hyperbolic set for it.